The rational decision structure of Baranwal's certified depth-error instance

Abstract

Baranwal (arXiv:2607.16676v2) certifies, for the pairwise message-passing classifier on the broadcast-labelled Poisson Galton–Watson tree with a two-point feature law at (Δ,γ,t₀)=(3,0.55,0.95), the first four values of the depth-error curve, E(0)=0.025, E(1)=0.02261938…, E(2) ∈ [0.03870641,0.03870642] and E(3) ∈ [0.07589622,0.07589623], by a never-renormalised lattice computation in which lattice points within 10⁻⁸ of a tie are counted as errors. We observe that at this instance every message is the logarithm of a rational number, 2artanh(γᵏ t₀)=log rₖ with rₖ=(1+γᵏ t₀)/(1-γᵏ t₀), so that the sign of the depth-ℓ statistic is an exact comparison of two integers. We derive the seven reduced ratios r₀,…,r₆, exhibit a distinguishing-prime certificate (29, 3433, 23, 59629, 41, 317) for their multiplicative independence—so the statistic never vanishes at depths ℓ ≤ 6 and the tie convention is never active there—verify exhaustively that no tie occurs at depth ≤ 2 on the box m₀= ± 1, |m₁| ≤ 30, |m₂| ≤ 70 (17,202 exact comparisons of integers of up to 367 digits), and identify the exact depth-one decision thresholds m₁ ≤ -4 (root sign +1) and m₁ ≤ 3 (root sign -1), the general half by induction, which express E(1) as a two-term Skellam sum. The two-point law has atoms, so at this instance the source's tie clause is not a null event a priori; the certificate shows it is vacuous. The ratios, the certificate, the exhaustive check, the threshold inequalities and five controls are machine-checked in Lean 4 with no axiom beyond propext and Quot.sound; the step from the certificate to independence is a one-line valuation argument. As uncertified evidence we report E(4) ∈ [0.10714,0.10719] and E(5) ∈ [0.13364,0.13375] from a scalar sub-probability lattice in floating point, and price a certified version.

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